The Economics of Exhaustible Resources
Author(s): Harold Hotelling
Source: Journal of Political Economy , Apr., 1931, Vol. 39, No. 2 (Apr., 1931), pp. 137175
Published by: The University of Chicago Press
Stable URL: https://www.jstor.org/stable/1822328
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THE JOURNAL OF
POLITICAL ECONOMY
Volume 39 APRIL 1931 Number 2
THE ECONOMICS OF EXHAUSTIBLE RESOURCES
I. THE PECULIAR PROBLEMS OF MINERAL WEALTH
C ONTEMPLATION of the world's disappearing supplies
of minerals, forests, and other exhaustible assets has
led to demands for regulation of their exploitation. The
feeling that these products are now too cheap for the good of
future generations, that they are being selfishly exploited at too
rapid a rate, and that in consequence of their excessive cheapness
they are being produced and consumed wastefully has given rise
to the conservation movement. The method ordinarily proposed
to stop the wholesale devastation of irreplaceable natural resources, or of natural resources replaceable only with difficulty
and long delay, is to forbid production at certain times and in
certain regions or to hamper production by insisting that obsolete
and inefficient methods be continued. The prohibitions against
oil and mineral development and cutting timber on certain government lands have this justification, as have also closed seasons
for fish and game and statutes forbidding certain highly efficient
means of catching fish. Taxation would be a more economic meth-
od than publicly ordained inefficiency in the case of purely commercial activities such as mining and fishing for profit, if not also
for sport fishing. However, the opposition of those who are making the profits, with the apathy of everyone else, is usually suffi-
cient to prevent the diversion into the public treasury of any conI37
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I38 HAROLD HOTELLING
siderable part of the proceeds of the exploitation of natural resources.
In contrast to the conservationist belief that a too rapid exploi-
tation of natural resources is taking place, we have the retarding
influence of monopolies and combinations, whose growth in industries directly concerned with the exploitation of irreplaceable
resources has been striking. If "combinations in restraint of
trade" extort high prices from consumers and restrict production,
can it be said that their products are too cheap and are being sold
too rapidly?
It may seem that the exploitation of an exhaustible natural
resource can never be too slow for the public good. For every pro-
posed rate of production there will doubtless be some to point to
the ultimate exhaustion which that rate will entail, and to urge
more delay. But if it is agreed that the total supply is not to be
reserved for our remote descendants and that there is an optimum
rate of present production, then the tendency of monopoly and
partial monopoly is to keep production below the optimum rate
and to exact excessive prices from consumers. The conservation
movement, in so far as it aims at absolute prohibitions rather
than taxation or regulation in the interest of efficiency, maybe
accused of playing into the hands of those who are interested in
maintaining high prices for the sake of their own pockets rather
than of posterity. On the other hand, certain technical conditions
most pronounced in the oil industry lead to great wastes of material and to expensive competitive drilling, losses which may be
reduced by systems of control which involve delay in production.
The government of the United States under the present administration has withdrawn oil lands from entry in order to conserve
this asset, and has also taken steps toward prosecuting a group
of California oil companies for conspiring to maintain unduly
high prices, thus restricting production. Though these moves
may at first sight appear contradictory in intent, they are really
aimed at two distinct evils, a Scylla and Charybdis between which
public policy must be steered.
In addition to these public questions, the economics of exhaus-
tible assets presents a whole forest of intriguing problems. The
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES I39
static-equilibrium type of economic theory which is now so well
developed is plainly inadequate for an industry in which the in-
definite maintenance of a steady rate of production is a physical
impossibility, and which is therefore bound to decline. How much
of the proceeds of a mine should be reckoned as income, and how
much as return of capital? What is the value of a mine when its
contents are supposedly fully known, and what is the effect of
uncertainty of estimate? If a mine-owner produces too rapidly,
he will depress the price, perhaps to zero. If he produces too slowly, his profits, though larger, may be postponed farther into the
future than the rate of interest warrants. Where is his golden
mean? And how does this most profitable rate of production vary
as exhaustion approaches? Is it more profitable to complete the
extraction within a finite time, to extend it indefinitely in such a
way that the amount remaining in the mine approaches zero as a
limit, or to exploit so slowly that mining operations will not only
continue at a diminishing rate forever but leave an amount in
the ground which does not approach zero? Suppose the mine is
publicly owned. How should exploitation take place for the great-
est general good, and how does a course having such an objective
compare with that of the profit-seeking entrepreneur? What of
the plight of laborers and of subsidiary industries when a mine is
exhausted? How can the state, by regulation or taxation, induce
the mine-owner to adopt a schedule of production more in harmony with the public good? What about import duties on coal
and oil? And for these dynamical systems what becomes of the
classic theories of monopoly, duopoly, and free competition?
Problems of exhaustible assets are peculiarly liable to become
entangled with the infinite. Not only is there infinite time to consider, but also the possibility that for a necessity the price might
increase without limit as the supply vanishes. If we are not to
have property of infinite value, we must, in choosing empirical
forms for cost and demand curves, take precautions to avoid assumptions, perfectly natural in static problems, which lead to
such conditions.
While a complete study of the subject would include semireplaceable assets such as forests and stocks of fish, ranging grad-
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I40 HAROLD HOTELLING
ually downward to such short-time operations as crop carry-
overs, this paper will be confined in scope to absolutely irreplaceable assets. The forests of a continent occupied by a new
population may, for purposes of a first approximation at least,
be regarded as composed of two parts, of which one will be replaced after cutting and the other will be consumed without re-
placement. The first part obeys the laws of static theory; the
second, those of the economics of exhaustible assets. Wild life
which may replenish itself if not too rapidly exploited presents
questions of a different type.
Problems of exhaustible assets cannot avoid the calculus of
variations, including even the most recent researches in this
branch of mathematics. However, elementary methods will be
sufficient to bring out, in the next few pages, some of the principles of mine economics, with the help of various simplifying as-
sumptions. These will later be generalized in considering a series
of cases taking on gradually some of the complexities of the actual
situation. We shall assume always that the owner of an exhaustible supply wishes to make the present value of all his future prof-
its a maximum. The force of interest will be denoted by y, so
that emit is the present value of a unit of profit to be obtained
after time t, interest rates being assumed to remain unchanged
in the meantime. The case of variable interest rates gives rise to
fairly obvious modifications.,
2. FREE COMPETITION
Since it is a matter of indifference to the owner
er he receives for a unit of his product a price
poeYt after time t, it is not unreasonable to expec
will be a function of the time of the form p =
apply to monopoly, where the form of the demand function is
bound to affect the rate of production, but is characteristic of
completely free competition. The various units of the mineral
are then to be thought of as being at any time all equally valuable,
excepting for varying costs of placing them upon the market.
' As in "A General Mathematical Theory of Depreciation," by Harold Hotelling,
Journal of the American Statistical Association, September, I925.
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES I4I
They will be removed and used in order of accessibility, the most
cheaply available first. If interest rates or degrees of impatience
vary among the mine-owners, this fact will also affect the order
of extraction. Here p is to be interpreted as the net price received
after paying the cost of extraction and placing upon the marketa convention to which we shall adhere throughout.
The formula
(I)
P=poeyt
fixes the relative prices at different times under free competition.
The absolute level, or the value po of the price when t = o, will depend upon demand and upon the total supply of the substance.
Denoting the latter by a, and putting
q =ft(pt)
for the quantity taken at time t if the price is p, we have the
equation,
T
T
(2) fq dt=f(poeyt, t)dt= a,
the upper limit T being the time of final exhaustion. Since q will
then be zero, we shall have the equation
(3) f(poe-1 , T)= o
to determine T.
The nature of these solutions will depend upon the function
f (p, t), which gives q. In accordance with the usual assumptions,
we shall assume that it is a diminishing function of p, and depends
upon the time, if at all, in so simple a fashion that the equations
all have unique solutions.
Suppose, for example, that the demand function is given by
q=5-Px (o<P<5)
q=o for P25,
independently of the time.
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I42 HAROLD HOTELLING
As q diminishes and approaches zero, p increases toward the
value 5, which represents the highest price anyone will pay.
Thus at time T,
poeYT = 5
The relation (2) between the unknowns p0 and T becomes in this
case
a=f (5-poeY t)dt=5T-po(eYT,-i)fy.
Eliminating p0, we have
a/5= T+ (e-'Y1-i )1-y
that is,
e YT= I -Y(a/5-T)
Now, if we plot as functions of T
y1=e-YT
and
y2 =I+-Y(a15-T),
we have a diminishing exponential curve whose slope where it
crosses the y-axis is -y, and a straight line with the same slope.
The line crosses the y-axis at a higher point than the curve, since,
when T = o, Y. < Y2. Hence there is one and only one positive value
of T for which Y. = Y2. This value of T gives the time of complete
exhaustion. Clearly it is finite.
If the demand curve is fixed, the question whether the time
until exhaustion will be finite or infinite turns upon whether a
finite or infinite value of p will be required to make q vanish. For
the demand function q = e-P, where b is a constant, the exploitation will continue forever, though of course at a gradually dimin-
ishing rate. If q = a - fp, all will be exhausted in a finite time.
In general, the higher the price anticipated when the rate of pro-
duction becomes extremely small, compared with the price for a
more rapid production, the more protracted will be the period of
operation.
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES I43
3. MAXIMUM SOCIAL VALUE AND STATE INTERFERENCE
As in the static case, there is under free competition in
sence of complicating factors a certain tendency toward maximizing what might be called the "total utility" but is better called
the "social value of the resource." For a unit of time this quantity may be defined as
(4) u(q) p(q) dq,
where the integrand is a dimini
is the quantity actually placed upon the market and consumed.
If future enjoyment be discounted with force of interest y, the
present value is
AT
V= u[q(t)]e-Yt dt.
Since q dt is fixed, the production schedule q (t) which ma
V a maximum must be such that a unit increment in q will incr
the integrand as much at one time as at another. That is,
d u[q(t)16-71 ,
which by (4) equals pe-"t, is to be a constant. Calling this constant Pot we have
p=poe'ty
the result (i) obtained in considering free competition. That this
gives a genuine maximum appears from the fact that the second
derivative is essentially negative, owing to the downward slope
of the demand curve.
This conclusion does not, of course, supply any more justification for laissez faire with the exploitation of natural resources than
with other pursuits. It shows that the true basis of the conservation movement is not in any tendency inherent in competition
under these ideal conditions. However, there are in extractive
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I44 HAROLD HOTELLING
industries discrepancies from our assumed conditions leading to
particularly wasteful forms of exploitation which might well be
regulated in the public interest. We have tacitly assumed all the
conditions fully known. Great wastes arise from the suddenness
and unexpectedness of mineral discoveries, leading to wild rushes,
immensely wasteful socially, to get hold of valuable property.
Of this character is the drilling of "offset wells" along each side
of a property line over a newly discovered oil pool. Each owner
must drill and get the precious oil quickly, for otherwise his
neighbors will get it all. Consequently great forests of tall der-
ricks rise overnight at a cost of $50,000 or more each; whereas a
much smaller number and a slower exploitation would be more
economic. Incidentally, great volumes of natural gas and oil are
lost because the suddenness of development makes adequate storage impossible.2
The unexpectedness of mineral discoveries provides another
reason than wastefulness for governmental control and for special
taxation. Great profits of a thoroughly adventitious character
arise in connection with mineral discoveries, and it is not good
public policy to allow such profits to remain in private hands. Of
course the prospector may be said to have earned his reward by
effort and risk; but can this be said of the landowner who discovers the value of his subsoil purely by observing the results of
his neighbors' mining and drilling?
The market rate of interest y must be used by an entrepreneur
in his calculations, but should it be used in determinations of
q
social value and optimum public policy? The use of pdq as a
measure of social value in a unit of time, whereas the smaller
quantity pq would be the greatest possible profit to an owner for
the same extraction of material, suggests that a similar integral
be used in connection with the various rates of time-preference.
There is, however, an important difference between the two cases
in that the rate of interest is set by a great variety of forces,
chiefly independent of the particular commodity and industry
in question, and is not greatly affected by variations in the out2 Cf. George W. Stocking, The Oil Industry and the Competitive System, Hart
Schaffner and Marx prize essay (Houghton Mifflin, I928).
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES I45
put of the mine or oil well in question. It is likely, therefore, that
in deciding questions of public policy relative to exhaustible resources, no large errors will be made by using the market rate of
in teres. Of course, changes in this rate are to be anticipated,
especially in considering the remote future. If we look ahead to
a distant time when all the resources of the earth will be near
exhaustion, and the human race reduced to complete poverty,
we may expect very high interest rates indeed. But the exhaustion of one or a few types of resources will not bring about this
condition.
The discounting of future values of u may be challenged on the
ground that future pleasures are ethically equivalent to present
pleasure of the same intensity. The reply to this is that capital
is productive, that future pleasures are uncertain in a degree increasing with their remoteness in time, and that V and u are
concrete quantities, not symbols for pleasure. They measure the
social value of the mine in the sense concerned with the total production of goods, but not properly its utility or the happiness to
which it leads, since this depends upon the distribution of wealth,
and is greater if the products of the mine benefit chiefly the poor
than if they become articles of luxury. A platinum mine is of
greater general utility when platinum is used for electrical and
chemical purposes than when it is pre-empted by the jewelry
trade. However, we must leave questions of distribution of wealth
to be dealt with otherwise, perhaps by graded income and in-
heritance taxes, and consider the effects of various schedules of
operation upon the total value of goods produced. It is for this
reason that we are concerned with V.
The general question of how much of its income a people should
save has been beautifully treated by F. P. Ramsey.3
Money metals, of course, occasion very special cause for pub-
lic concern. Not only does gold production tend to unstabilize
prices; but if the uses in the arts can be neglected, the costs of
discovery, extraction, and transportation from the mine are, from
the social standpoint, wasted.
Still a different reason for caution in deducing a laissez faire
3 "A Mathematical Theory of Saving," Economic Journal, XXXVIII (I928),
543.
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I46 HAROLD HOTELLING
policy from the theoretical maximizing of V under "free" compe-
tition is that the actual conditions, even when competition exists,
are likely to be far removed from the ideal state we have been
postulating. A large producing company can very commonly af-
fect the price by varying its rate of marketing. There is then
something of the monopoly element, with a tendency toward un-
due retardation of production and elevation of price. This will
be considered further in our last section. The monopoly problem
FIG. I. y = pq. The tangent turns counterclockwise. The value of the mine is
proportional to the distance to 0 from the intersection of the tangent with the qaxis.
of course extends also to non-extractive industries; but in dealing
with exhaustible resources there are some features of special interest, which will now be examined.
4. MONOPOLY
The usual theory of monopoly prices deals with the maximum
point of the curve
y=pq,
y being plotted as a function either of p or of q, each of these variables being a diminishing function of the other (Fig. i). We now
consider the problem of choosing q as a function of t, subject to
the condition
(00
(5)
q
d=a
a,
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES I47
so as to maximize the present value,
(6) J=fqp(q)e--t dt,
of the profits of the owner of a mine. We do not restrict q to be
a continuous function of t, though p will be considered a continuous function of q with a continuous first derivative which is
nowhere positive. The upper limit of the integrals may be taken
as Xo even if the exploitation is to take place only for a finite time
T7 for then q=o when t > T.
This may or may not be considered a problem in the calculus
of variations; some definitions of that subject would exclude our
problem because no derivative is involved under the integral
signs, though the methods of the science may be applied to it.
However, the problem may be treated fairly simply by observing
that
(7) q p(q)e--X Xq,
where X is a Lagrange multiplier, is to be a maximum for every
value of t. We must therefore have
(8) e et d- (pq)-X=o,
dq
and also
(9) east d 2 (Pq) <o
Evidently (8) may also be written
(IO) Yt = d(p?) =P+q P=e t
dq ~~dq
the contrast with the competitive conditions of the last section
appearing in the term q dp/dq.
The constant X is determined by solving (8) or (Io) for q as a
function of X and t and substituting in (5). Upon integrating from
o to T an equation will then be obtained for X in terms of T and of
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I48 HAROLD HOTELLING
the amount a initially in the mine, which is here assumed to be
known. The additional equation required to determine T is obtained by putting q = o for I = T.
In general, if p takes on a finite value K as q approaches zero,
q dp/dq also remaining finite, (8) or (io) can be written
dq
_ q)Key(t -T)
Suppose, for example, that the demand function is
p = (I - e-Kq)lq
= K -K2 ql2!+K3q 2/3 !-....
where K is a positive constant. For every positive value of q this
expression is positive and has a negative derivative. As q approaches zero, p approaches K. We have
y=Pq=i--e-KQ
y'= Ke-Kq = Xe'yt
whence
q =(log K/A - yt)IK,
this expression holding when t is less than T, the time of ultimate
exhaustion. When I= T, q is of course zero. We have, therefore,
putting q = o for t = T,
log K/X= yT;
and from (5)
rT
T
a= (log KI - -t) dtIK = y (T-t)dtIK
= yT2/2K
so that
T=1/2Ka/ly,
log K/X=1l/2K-ya,
giving finally,
q= 7(112Ka1-y-t)/1K.
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES 149
5. GRAPHICAL STUDY: DISCONTINUOUS SOLUTIONS
The interpretation of (IO) in terms of Figure I is that the rate
of production is the abscissa of the point of tangency of a tangent
line which rotates counterclockwise. The slope of this line is
proportional to a sum increasing at compound interest.
Other graphical representations of the exhaustion of natural
resources are possible. Drawing a curve giving y'=d(pq)/dq as
a function of q (Fig. 2), we have for the most profitable rate of
Iy
X ~~CA
FIG. 2. RS rises with increasing speed. Its length is the rate of production
and diminishes.
extraction the length of a horizontal line RS which rises like compound interest.
The waviness with which these curves have been drawn sug-
gests that the solution obtained in this way is not unambiguous.
Such waviness will arise if the demand function is, for example
(II)
p=b-(q-I)3,
the derivative of which,
-3(q-I)2
is never positive. Here b is a constant, taken as I for Figure 2.
For this demand function
.! Lb / .q-i_(q- I )2
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I50 HAROLD HOTELLING
When the rising line RS reaches the position AC, the point S
whose abscissa represents the rate of production might apparently continue along the curve to B and then jump to D; or it might
jump from A to C and then move on through D; or it might leave
the arc AB at a point between A and B. At first sight there
would seem to be another possibility, namely, to jump from A to
C, to move up the curve to B, and then to leap to D. But this
would mean increasing production for a period. This is never so
profitable as to run through the same set of values of q in reverse
order, for the total profit would be the same but would be re-
ceived on the average more quickly if the most rapid production
takes place at the beginning of the period. Hence we may regard
q as always diminishing, though in this case with a discontinuity.
The values of q between which the leap is made in this case will
be determined in ? io; it will be shown that the maximum profit
will be reached if the monopolist moves horizontally from a certain point F on AB to a point G on CD.
6. VALUE OF A MINE MONOPOLY
To find the present value
=
td
of the profits which are to be realized in any interval t4 to t2
during which the maximizing value of q is a continuous function
of I, we integrate by parts:
pge t2 2d~t d(pq) dq
= _ J + I dy dq dt ent dt.
When we put
(I
2)
y=pq
and apply (Io), the last integral takes a simple form admitting
direct integration. This gives, after applying (io) also to eliminate e-et from the first term,
/
\
~~~~~t.
\/
'
t2
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES I5I
Now upon differentiating (I 2), we find
qy =y+q2 d2
dq*
Hence (13) can be written
(I4)
jt2
X
q2
dpIt
Nt,
yy
dq
t.
The expressions (13) and (14) provide very convenient means
of computing the discounted profits. Their validity will be shown
in ? IO to extend to cases in which q is discontinuous.
The expression
q-y/y'
which appears in (13) is, in terms of Figure i, the difference be-
tween the abscissa and the subtangent of a point on the curve.
It therefore equals the distance to the left of the origin of the
point where a tangent to the curve meets the q-axis.
The value of the mine when t=o is, in this notation, J. It
is 'X/' times the distance from the origin to the point of intersection with the negative x-axis of the initial tangent to the curve
of monopoly profit.
7. RETARDATION OF PRODUCTION UNDER MONOPOLY
Although the rate of production may suffer discontinuities
in spite of the demand function having a continuous derivative,
these breaks will always occur during actual production, never at
the end. Eventually q will trail off in a continuous fashion to
zero. This means that the highest point of the curve of Figure 2
corresponds to q o. To prove this, we use the monotonic decreasing character of p as a function of q, which shows that
y'(q) = P(q) +qp'(q)
is, for positive values of q, less than p(q), and that this in turn is
less than p(o). Hence the curve rises higher at the y'-axis than
for any level maximum at the right.
The duration of monopolistic exploitation is finite or infinite
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I52 HAROLD HOTELLING
according as y' takes on a finite or an infinite value when q approaches zero. This condition is a little different from that under
competition, where a finite value of p as q approaches zero was
found to be necessary and sufficient for a finite time. The two
conditions agree unless p remains finite while qp'(q) becomes
infinite, in which case the demand curve reaches the p-axis and is
tangent to it with contact of order higher than the first. In such
a case the period of operation is finite under competition but
infinite under monopoly. That this apparently exceptional case
is quite likely to exist in fact is indicated by a study of the gen-
eral properties of supply and demand functions applying the the-
ory of frequency curves, a fascinating subject for which space
will not be taken in this paper.
Such a study indicates that very high order contact of the de-
mand curve with the p-axis is to be expected, and therefore that
monopolistic exploitation of an exhaustible asset is likely to be
protracted immensely longer than competition would bring about
or a maximizing of social value would require. This is simply a
part of the general tendency for production to be retarded under
monopoly.
8. CUMULATED PRODUCTION AFFECTING PRICE
The net price p per unit of product received by the owner of a
mine depends not only on the current rate of production but also
on past production. The accumulated production affects both
cost and demand. The cost of extraction increases as the mine
goes deeper; and durable substances, such as gold and diamonds,
by their accumulation influence the market. In considering this
effect, the calculus of variations cannot be avoided; the following
formulation in terms of this science will include as special cases
the situations previously treated.
Let x be the amount which has been extracted from a mine,
q=dx/dt the current rate of production, and a the amount originally in the mine. Then p is a function of x as well as of q and t.
The discounted profit at time t= o, which equals the value of the
mine at that time, is
W) P~rx, q, .~e'yI
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES I53
If exhaustion is to come at a finite time T, we may
q = o for t> T, so that T becomes the upper limit. W
f(X, q, t)=pqe-t .
Then the owner of the mine (who is now assumed to have a
monopoly) cannot do better than to adjust his production so that
af d 0f
Ox dt 4q
In case f does not involve x, the first term is zero and the former
case of monopoly is obtained.
In general the differential equation is of the second order in x,
since q = dx/dt, and so requires two terminal conditions. One of
these is x = o for t = o. The other end of the curve giving x as a
function of t may be anywhere on the line x = a, or the curve may
have this line as an asymptote. This indefiniteness will be settled
by invoking again the condition that the discounted profit is a
maximum. The "transversality condition" thus obtained,
f-qf
that is,
Op
q2 49-=0,
clq
is equivalent to the proposition that, if p always diminishes when
q increases, the curve is tangent or asymptotic to the line x =a.
Thus ultimately q descends continuously to zero.
Suppose, for example, that q, x, and t all affect the net price
linearly. Thus
P=a-1q-cx+gt.
Ordinarily a, f3, and c will be positive, but g may ha
The growth of population and the rising prices to consumers of
competing exhaustible goods would lead to a positive value of g.
On the other hand, the progress of science might lead to the gradThis content downloaded from
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I54 HAROLD HOTELLING
ual introduction of new substitutes for the commodity in question,
tending to make g negative. The exhaustion of complementary
commodities would also tend toward a negative value of g.
The differential equation reduces, for this linear demand function, to the linear form
d2x dx
23dt2 2 dt j1--x-yt--y
Since 3, c, and y are positive, the root
are real and of opposite signs. Let m denote the positive and -n
the negative root. Since
mn-n=y,
ni9 is numerically greater than n. The solution is
x=Aem'+Be-n + #1 C - 2 gIC2- glcy+ a/c
whence
q=Amemt-Bn-nt+g/c.
Since x=o when I=o,
A+B- 20gIC2 -g/cy+a/c=o.
Since x = a and q = o at the time T of ultimate exhaustion,
AemT +Be-nT +gT/c - 20gIC2 -g/cy +a/c -a = o,
AmemT _Bne-nT+g/c =0.
From these equations A and B are eliminated by equating to zero
the determinant of their coefficients and of the terms not con-
taining A or B. After multiplying the first column by e-mT and
the second by enT, this gives:
e-mT enT - 20gIC2 - glc+a/c
mI I gT-c- 2ngC2 g/c-y+a/c-a =o.
m
-
n
g1C
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES I55
Expanding and using the relations mr-n =y, and mn
we have for /v and its derivative with respect to T,
/ = (e-mT -enT)g/c+ (ne-mT+ menT) (gT/c- 2#gC2-g/c+ a/Cc-a)
+ (m+n)(20g1C2+g1Cy - a/c'
1= (enT-emT )[T -i/+ (a-ac)/glg7/2
the last expression being useful in applying Newton's method to
find T. Obviously, the derivative changes sign for only one
value of T; for this value A has a minimum if g is positive, a
maximum if g is negative.
We may measure time in such units that Ay, the force of interest,
is unity. If money is worth 4 per cent, compounded quarterly,
the unit of time will then be about 25 years and i month.
With this convention let us consider an example in which there
is an upward secular trend in the price consumers are willing to
pay: take a=ioo, =I,c=4,g=i6, anda=io. Thenetamount
received per unit is in this case
P=Ioo-q-4x+I-6t.
Substituting the values of the constants, and noting that m= 2
and n= i, we have
e-2T eT I9
A = I I 4T+9 =(8T+I4)eT+(4T+I3)e 2T_ 57,
2
-I
4
Y= (eT-e-2T)(8T+ 22)
Evidently A < o for T = o, A= + oo for T = oo, and A'>o for all
positive values of T. Hence A= o has one and only one positive
root.
For the trial value T= i we have
A=5.IO, Al -77.5
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I56 HAROLD HOTELLING
Applying to T the correction - /A'= - .07 roughly, we take
T=.g3 as a second approximation. For this value of T,
whence -//A'= ooi.
The most profitable schedule of extraction will therefore exhaust the mine in about 0.93I unit of time, or about 23 years and
4 months, perhaps a surprisingly short time in view of the prospect of obtaining an indefinitely higher price in the future, at the
rate of increase of i6 per unit of time.
In order that the time of working a mine be infinite, it is necessary not only that the price shall increase indefinitely but that
it shall ultimately increase at least as fast as compound interest.
The last two equations for determining A and B now become,
since e2T 6.4366 and e-T .3942,
6.4366A + .3942B+-I2. 724o0
12.8732A-. 3942B+4 =0.
Hence A= -.866, B= -i8.I3; so that
x =-. 866e2t-i8. I3e t+44t+ I9 .
As a check we observe that this expression for x vanishes when I
=0.
Differentiating, we have
q=-I . 732e2t+I8. I3e-t+4,
showing how the rate of production begins at 20.40 and gradually
declines to zero. Substitution in the assumed expression for the
net price gives
p= Ioo-q-4X+i6t
= 20+5 . ig6e2t+54.39e-t,
showing a decline from 79.60 at the beginning to 74.90 at exhaus-
tion, owing to the greater cost of extracting the deeper parts of
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES I57
the deposit. The buyer of course pays an increasing, not a decreasing price, namely,
P+4X= ioo-q+i6t
=96+I. 732e2t- 8. I3e t+I6t.
This increases from 79.60 to II4.90.
9. THE OPTIMUM COURSE
To examine the course of exploitation of a mine which would
be best socially, in contrast with the schedule which a wellinformed but entirely selfish owner would adopt, we generalize
the considerations of ? 3. Instead of the rate of profit pq, we must
now deal with the social return per unit of time,
u= p(x, q, t)dq,
x and t being held constant in the integration. Taking again the
market rate of interest as the appropriate discount factor for
future enjoyments, we set
F = ue-alY
and inquire what curve of exploitation will make the total discounted social value,
V= Fdt
a maximum.
The characteristic equation
aF d aF
ax dt aq
reduces to
op d2x+?p dx p ou 3p
aq dt2 ax dt ox at
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I58 HAROLD HOTELLING
The initial condition is x = o for t = o. The other end-point of the
curve is movable on the line x - a = o, a being the amount originally in the mine. The transversality condition,
OF
F-q a=o,
reduces to
u-pq=o.
This is satisfied only for q=o, for otherwise we should have the
equation
p=IXp dq ,
stating that the ultimate price is the mean of the potential prices
corresponding to lower values of q. Since p is assumed to de-
crease when q increases, this is impossible. Even if 9p/Oq is zero
in isolated points, the equation will be impossible if, as is always
held, this derivative is elsewhere negative. Hence q=o at the
time of exhaustion.
If, as in ? 8, we suppose the demand function linear,
P=a-/3q-cx+gt,
the characteristic equation becomes
d2X dx
_ 0'-SY -d--Cyx= _9't+g-a-y
This differs from the corresponding equation for monopoly only
in that : is here replaced by f/2. In a sense, this means that the
decline of price, or marginal utility, with increase of supply counts
just twice as much in affecting the rate of production, when this is
in the control of a monopolist, as the public welfare would warrant.
The analysis of ? 8 may be applied to this case without any
qualitative change. The values of en and n depend on A, and are
therefore changed. The time T until ultimate exhaustion will be
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES I59
reduced, if social value rather than monopoly profit is to be maximized. For the numerical example given, T was found to be
0.93I unit of time under monopoly. Repeating the calculation
for the case in which maximum social value is the goal, we find
as the best value only 0.674I unit of time.
For different values of the constants, even with a linear de-
mand function, the mathematics may be less simple. For example, the equation A = o may have two positive roots instead of one.
This will be the case if the numerical illustration chosen be varied
by supposing that the sign of g is reversed, owing to the progres-
sive discovery of substitutes, the direct effect of passage of time
being then to decrease instead of increase the price. In such cases
a further examination is necessary of the two possible curves of
development, to determine which will yield a greater monopoly
profit or total discounted social value, according to our object.
IO. DISCONTINUOUS SOLUTIONS
Even if the rate of production q has a discontinuity, as in the
example of ? 5, the condition that ff dt shall be a maximum requires that each of the quantities
aq
'
q
must nevertheless be continuous.4 This will be true whether f
stands for discounted monopoly profit or discounted total utility.
The equation (8) on p. I47, may be written
aq
which shows, since the left-hand member is continuous, that X
must have the same value before and after the discontinuity.
When p is a function of q alone, the two continuous quantities
may be written in the notation of ? 4, yfe-'t and (y-qy') e-y
4 C. Caratheodory, "Uber die diskontinuirlichen LUisungen in der Variations-
rechnung," thesis, Gdttingen, I904, p. ii. The condition that the first of these quantities must be continuous is given in the textbooks, but for some reason the second
is generally omitted.
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i6o HAROLD HOTELLING
which shows that y' and y - qy' are continuous. Thus the expres-
sion X (q-y/y') appearing in (13), p. I50, is continuous. Conse-
quently the expressions (I3) or (N') pertaining to the differe
time-intervals may simply be added to obtain an expression of
the same form. Hence the present value of the discounted future
profits of the mine-and therefore of the mine-is in such cases
the difference between the values of
X(q-y/y')/'y
at present and at the time of exhaustion.
We are now ready to answer such questions as that raised at
the end of ? 5 as to the location of the discontinuity there shown
to exist in the most profitable schedule of production when the
demand function is
p=b-(q-i)3.
Since in this case
f= pqe-^ t= [bq-q(q-I )3]e-' At
the two quantities
b -(4q -i) (q -I)2,
3q2(q )2,
are continuous. Consequently
(4q-I) (q 1)2
and
q2(q- I)2
are continuous. If q1 denote the rate of production just before the
sudden jump and q2 the initial rate after it, this means that
(4 I -I) (q - I)2 =(4q2- i)(q2- I)2
q12(qx -1)2 = q22(q2 -I)2
The only admissible solution is:
qI=(3+1/3)/41. Ii830, q2=(3-V3)/4=-03i699.
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES i6i
I I. TESTS FOR A TRUE MAXIMUM
The equations which have been given for finding the production schedule of maximum profit or social value are necessary,
not sufficient, conditions for maxima, like the vanishing of the
first derivative in the differential calculus. We must also consider
more definitive tests.
The integrals which have arisen in the problems of exhaustible
assets are to be maxima, not necessarily for the most general
type of variation conceivable for a curve, but only for the socalled "special weak" variations. The nature of the economic sit-
uation seems to preclude all variations which involve turning
time backward, increasing the rate of production, maintaining
two different rates of production at the same time, or varying
production with infinite rapidity. Extremely sudden increases in
production usually involve special costs which will be borne only
under unexpected conditions, and are to be avoided in long-term
planning. Likewise sudden decreases involve social losses of great
magnitude such as unemployment, which even a selfish monopolist will often try to prevent. This will be considered further in
the next section. It is indeed possible that in some special cases
these "strong" variations might take on some economic significance, but such a situation would involve forces of a different sort
from those with which economic theory is ordinarily concerned.
The critical tests which must be applied are by the foregoing
considerations reduced to two-those of Legendre and Jacobi.S
The Legendre test requires, in order that the total discounted
utility or social value (? 9) shall be a maximum, that
a2u a (p
aqq2 aq<
a condition which is always held to obtain save in exceptional
cases. In order that the chosen curve shall yield a genuine maximum for a monopolist's profit, the Legendre test requires that
a2(pq) 2 ap+ a2p<
aq2 =2q aq~2<
5 A. R. Forsyth, Calculus of Variations (Cambridge, 1927), pp. I7-28.
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I62 HAROLD HOTELLING
This means that the curve of Figure i is convex upward at all
points touched by the turning tangent. The re-entrant portions,
if any, are passed over, producing discontinuities in the rate of
production.
When the solution of the characteristic equation has been
found in the form
x=O(t, A, B),
A and B being arbitrary constants, the Jacobi test requires that
af/aA
shall not take the same value for two different values of t. For the
example of ? 8 this critical quantity is simply e(m+n)t, which obviously satisfies the test. The solution represents a real, not an
illusory maximum for the monopolist's profit. The like is true for
the schedule of production maximizing the total discounted utility with the same demand function. Each case must, however,
be examined separately, as the test might show in some instances
that a seeming maximum could be improved.
I2. THE NEED FOR STEADINESS IN PRODUCTION
The demand function giving p may involve not only the rate of
production q, but also the rate of change q' of q. Such a condition
would display a duality with that considered by C. F. Roos6 and
G. C. Evans,7 who hold that the quantity of a commodity which
can be sold per unit of time depends ordinarily upon the rate of
change of the price, as well as upon the price itself. If p is a function of x, q, q', and t, the maximum of monopoly profit or of social
value can only be obtained if the course of exploitation satisfies
a fourth-order differential equation.
6 "A Pynamical Theory of Economics," Journal of Political Economy, XXXV
(I927), 632, and references there given; also "A Mathematical Theory of Depreciation and Replacement," American Journal of Mathematics, L (1928), 147.
7 "The Dynamics of Monopoly," American Mathematical Monthly, Vol. XXXI
(1924); also Mathematical Introduction to Economics (McGraw-Hill Book Co., 1930).
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES i63
More generally we might suppose that p and its rate of change
p' are connected with x, q, q', and t by a relation
<(p, p', x, q, q'I t)=o
This presents a Lagrange problem, which can be dealt with by
known methods.8 A further generalization is to suppose that the
price, the quantity, and their derivatives are subject to a relation
in the nature of a demand function which also involves an integral or integrals giving the effect of past prices and rates of consumption.9
Capital investment in developing the mine and industries essential to it is a source of a need for steady production; the desirability of regular employment for labor is another. Under the
term "capital" might possibly be included the costs, both to
employers and to laborers, in drawing laborers to the mine from
other places and occupations. The returning of these laborers to
other occupations as production declines would have to be reckoned as part of the social cost. Whether this would enter into
the mine-owner's costs would probably depend upon whether the
laborers have at the beginning sufficient information and bargaining power to insist upon compensation for the cost to them of the
return shift.
Problems in which the fixity of capital investment plays a part
in determining production schedules may be dealt with by intro-
ducing new variables x1, X2,. ,.. to represent the various types
of capital investment involved. In so far as these variables are
continuous, the problem is that of maximizing an integral involv-
ing x, XI, X2 .... and their derivatives, using well-known
methods. The simultaneous equations
ajf dt df =0 o, I 2aj
axi dt dx0 (io I, 2. II; x0=x; X~=dxi/ dt)
8 Cf. G. A. Bliss, "The Problem of Lagrange in the Calculus of Variations"
(mimeographed by 0. E. Brown), (University of Chicago Bookstore).
9 See "Generalized Lagrange Problems in the Calculus of Variations," by C. F.
Roos, Transactions of the American Mathematical Society, XXX, (I927), 360.
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i64 HAROLD HOTELLING
are necessary for a maximum. The depreciation of mining equipment raises considerations of this kind.
The cases considered in the earlier part of this paper all led to
solutions in which the rate of production of a mine always decreases. By considering the influence of fixed investments and
the cost of accelerating production at the beginning, we may be
led to production curves which rise continuously from zero to a
maximum, and then fall more slowly as exhaustion approaches.
Certain production curves of this type have been found statistically to exist for whole industries of the extractive type, such
as petroleum production
I3. CAPITAL VALUE TAXES AND SEVERANCE TAXES
An unanticipated tax upon the value of a mine will have no
effect other than to transfer to the government treasury a part
of the mine-owner's income. An anticipated tax at the rate a per
year and payable continuously will have the same effects upon
the value of the mine and the schedule of production as an increase of the force of interest by a. This we shall now prove.
From the income pq from the mine at time t must now be deducted the tax, aJ(t). Consequently the value at time -r is
J(T) = ;[pq-aJ(I)]e--a(t-T)dt
This integral equation in J reduces by differentiation to a differential equation:
J(T) =-pq+aJ(T) +'yJ (T)
The solution is found by well-known methods. The constant of
integration is evaluated by means of the condition that J(T) = o.
We have:
T
so that a is merely added to Ay.
Io C. E. Van Orstrand, "On the Empirical Representation of
Curves," Journal of the Washington Academy of Sciences, XV, (1
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES I65
Quite a different kind of levy is represented by the "severance
tax."", Such a tax, of so much per unit of material extracted from
the mine, tends to conservation. The ordinary theory of monopoly of an inexhaustible commodity suggests that the incidence of
such a tax is divided between monopolist and consumer, equally
in the case of a linear demand function. However, for an exhaus-
tible supply the division is in a different proportion, varying with
time and the supply remaining. Indeed, the imposition of the
tax will lead eventually to an actually lower price than as if there
had been no tax.
Consider the linear demand function
P=a-fq
and, for simplicity, no cost of production. The rate of net profit,
after paying a tax v per unit extracted, will be
(p-v)q= (a-v)q-#q
As in ? 4, the derivative increases as compound interest:
a-V-23q= Xeyt .
Since ultimately q = o and t = T, we obtain
II A variant is an ad valorem tax. A great deal of information and discussion concerning these taxes is contained in the biennial Report of the Minnesota State Tax
Commission, I928, (St. Paul). From p. iii of this report it appears that Alabama
since I927 has had a severance tax of 2' cents a ton on coal, 4' cents a ton on iron
ore, and 3 per cent on quarry products; Montana taxes coal extracted at 5 cents per
ton; Arkansas imposes a tax of 2 2 per cent on the gross value of all natural resources
except coal and timber, I per cent on coal, and 7 cents per i,ooo board feet on timber.
Minnesota taxes iron ore extracted at 6 per cent on value minus cost of labor and
materials used in mining, and also assesses ore lands at a higher rate than other prop-
erty for the general property tax. These taxes are not based entirely on the conservation idea, but aim also at taxing persons outside the state, or "retaining for the
state its natural heritage." Since Minnesota produces about two-thirds of the iron
ore of the United States, the outside incidence is doubtless accomplished. Mexican
petroleum taxes have the same object. The Minnesota Commission believes that
prospecting for ore has virtually ceased on account of the high taxes.
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i66 HAROLD HOTELLING
whence eliminating X and solving for q,
q = I-Oe(t - T) ](a-v)/2,#
The time of exhaustion T is related to the amount originally
in the mine through the equation
af= q dt = (-yT+ eyT - ) (ac-V)/2f-y
whence
dT203a (i-e-,yT)
d
dT(a-v)2
showing how much of an increase in time of exploitation is likely
to result from the imposition of a small severance tax. The effect
upon the rate of production at time t is
dqr = d- - dv+ -a- - d T
=dv{ - I+eY(tT)[I+2,2fya/(a-v) (i-e eyT)]}/23 .
From the form of the demand function it follows that the increase in price at time t is
dp= =-,dq = dv I -e-Y(t-T) [2+ #ya1(a-v) ( I-e- yT)] }
If a is very large, then so is T; the expression in curly brackets
will, for moderate values of t, differ infinitesimally from 2, reducing to the case of monopoly with unlimited supplies. However, dp
will always be less than ldv and, as exhaustion approaches, will
decline and become negative. Finally, when t = T, the price of
the tax-paid articles to buyers is lower by
f^yav/(a-v)(i -e-yT)
than the ultimate price if there had been no tax. The price will,
nevertheless, be so high that very little of the commodity will be
bought.
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES i67
A tax on a monopolist which will lead him to reduce his prices
is reminiscent of Edgeworth's paradox of a tax on first-class
railway tickets which makes the monopolistic (and unregulated)
owner's most profitable course the reduction of the prices both
of first- and of third-class tickets, besides paying the tax himself.'2 The case of a mine is, however, of a distinct species from
Edgeworth's, and cannot be assimilated to it by treating ore extracted at different times as different commodities. Indeed, in the
simple case of mine economics which we are now considering, the
demands at different times are not correlated; supplies put upon
the market now and in the future neither complement nor compete with each other. Correlated demand of a particular type was,
on the other hand, an essential feature of Edgeworth's phenomenon.
The ultimate lowering of price and the extension of the life of
a mine as a result of a severance tax are not peculiar to the linear
demand function, but hold similarly for any declining demand
function p(q) whose slope is always finite. This general proposition does not depend upon the tax being small.
The conclusion reached in the linear case that the division of
the incidence of the tax is more favorable to the consumer than
for an inexhaustible supply is probably true in general; at least
this is indicated by an examination of a number of demand
curves. However, the general proposition seems very difficult to
prove.
Since the severance tax postpones exhaustion, falls in considerable part on the monopolist, and leads ultimately to an actual
lowering of price, it would seem to be a good tax. It is particularly
to be commended if the monopolist is regarded as unfairly possessed of his property, and there is no other feasible means of
taking away from him so great a portion of it as the severance
tax will yield. However, the total wealth of the community may
be diminished rather than increased by such a tax. Considering
as in ? 3 the integral u of the prices p which buyers are willing to
pay for quantities below that actually put on the market, and
22 Economic Journal, VII (I897), 23I, and various passages in Edgeworth's
Papers Relating to Political Economy.
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i68 HAROLD HOTELLING
the time-integral U of values of u discounted for interest, we have
in the case of linear demand just discussed,
uf= (a - q)dq= aq- 'q2
If we were considering the portion of this social benefit which in-
ures to consumers, we should have to subtract the portion pq
which they pay to the monopolist, an amount from which he
would have to subtract the tax, which benefits the state. But the
sum of all these benefits is u, which is affected by the tax only
as this affects the rate of production q.
If, for simplicity, we measure time in such units that y = i, the
rate of production determined earlier in this section becomes
q= (I -et-T)(a-V)/2|3 .
Substituting this in the expression for u and the result in U, we
obtain
eT
U=ue-t dt
=(a-v)[4a(i-eT-Te T) -(a-v)(I-21eT-e e2T)] 8$
We differentiate U and then, to examine the effect of a small tax,
put v = o. The results simplify to:
d -=-(I-e- Ta/40
3v
au
a = [(T+I)e e--e T]a /40d
From the amount initially in the mine,
a= (T+e T-I)/2I3,
we obtain, as on page i66,
dT 2f3a
dv a2(I-e-T)
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES i69
when v =o. Substituting here the preceding value for a we find,
after simplification,
dU aU aU dT a eT+e-T-2-T2
d av aT dv 40 eT-I
The numerator of the last fraction may be expanded in a convergent series of powers of T in which all the terms are positive.
Hence dU/dv is negative.
Thus a small tax on a monopolized resource will diminish its
total social value, at least if the demand function is linear.
Whether this is true for demand functions in general is an un-
solved problem.
We have here supposed the tax v to be constant, permanent,
and fully foreseen. Since an unforeseen tax will have unforeseen
results, we can scarcely build up a general theory of such taxes.
However, any tax of amount varying with time in a manner definitely fixed upon in advance will have predictable results. In this
connection, an interesting problem is to fix upon a schedule of
taxation v, which may involve the rate of production q and the
cumulated production x, as well as the time, such that, when the
monopolist then chooses his schedule of production to maximize
his profit, the social value U will be greater than as if any other
tax schedule had been adopted. This leads to a problem of La-
grange type in the calculus of variations, one end-point being
variable. Putting q = dx/dt and
AT
T
J=f(x, q, v, t) dt, U=fF(x, q, I) di,
the problem is to choose v, subject to the differential relation
Of f aO _ d af +af av
ax vO ax dt \dq O vq/0'
so that U will be a maximum. In general, of course, a still greater
value of U would be obtainable, at least in theory, by public
ownership and operation.
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I70 HAROLD HOTELLING
I4. MINE INCOME AND DEPLETION
With income taxes we are not concerned except for the determination of the amount of the income from a mine. The problem
of allowance for depletion has been a perplexing one. It has been
said that if the value of ore removed from the ground could be
claimed as a deduction from income, then a mining company having no income except from the sale of ore could escape payment of
income tax entirely. The fallacy of this contention may be examined by considering the value of the mine at time t,
J(t)=j pqe-'(-T)dT.
In this integral p and q have the values corresponding to the
time r, later than t, assigned by whatever production schedule
has been adopted, whether this results from competition, from a
desire to maximize monopoly profit, or from any other set of
conditions. The net income consists of the return from sales of
material removed (cost of production and selling having as usual
been deducted), minus the decrease in the value of the mine.
It therefore equals, per unit of time,
pq+dJ/dt;
and from the expression for J, this is exactly -yJ. In other words,
any particular production schedule fixes the value of the mine at
such a figure that the income at any time, after allowing for depletion, is exactly equal to the interest on the value of the investment at that time.
But, although the rate of decline in value of a mine seems a
logical quantity to define as depletion and to deduct from income,
such is not the practice of income-tax administrations, at least
in the United States. The value of the property upon acquisition,
or on March I, I9I3, a date shortly before the inauguration of the
tax, if acquired before that time, is taken as a basis and divided by
the number of units of material estimated to have been in the
ground at that time. The resulting "unit of depletion," an
amount of money, is multiplied by the number of tons, pounds,
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES I7I
or ounces of material removed in a year to give the depletion for
that year. The total of depletion allowances must not exceed the
original value of the property.
The differences between the two methods of calculating de-
pletion arise from the uncertainties of valuation and of forecasting price, demand, production, costs, interest rates, and amount
of material remaining. If the theoretical method were applied, a
year in which the mine failed to operate would still be set down as
yielding an income equal to the interest on the investment value.
This seems anomalous only because of another defect, from the
theoretical standpoint, in income-tax laws: the non-taxation of
increase in value of a property until the sale of the property.
During a year of idleness, if foreseen, the value of the property
is actually increasing, for the idle year has been considered in fixing the value at the beginning of the year.
An amendment made to the United States federal income-tax
law in i9i8 provides that the valuation upon which depletion is
calculated may under certain circumstances be taken, not as the
value of the property when acquired or in I9I3, but the higher
value which it later took when its mineral content was discovered.
This provision has the effect of materially increasing depletion
allowances, and so of reducing tax payments. The sudden in-
crease in value when the mineral is discovered might well be regarded as taxable income, but is not so regarded by the law
unless the property is immediately sold. The framers of the statute seem indeed, according to its language, to have considered
this increase in value a reward for the efforts and risks of prospect-
ing, which would suggest that it is of the nature of income, a
reasonable position. However, the object of the amendment is to
treat this increment as pre-existing capital value, to be returned
to the owner by sale of the mineral. The amendment appears to
be inconsistent and quite too generous to the owners particularly
affected.
I 5. DUOPOLY
Intermediate between monopoly and perfect competition, and
more closely related than either to the real economic world, is
the condition in which there are a few competing sellers. In a for-
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I72 HAROLD HOTELLING
mer paper'3 this situation was discussed for the static case, with
special reference to a factor usually ignored, the existence with
reference to each seller of groups of buyers who have a special
advantage in dealing with him in spite of possible lower prices
elsewhere. More than one price in the same market is then possible, and with a sort of quasi-stability which sets a lower limit to
prices, as well as the known upper limit of monopoly price.
For exhaustible resources the corresponding problems of competition among a small number of entrepreneurs may be studied
in the first instance by means of the jointly stationary values of
the several integrals representing discounted profits. We need
not confine ourselves, as we have done for convenience in dealing
with monopoly, to a single mine for each competitor. Let there
be m competitors, and let the one numbered i control ni mines,
whose production rates and initial contents we shall denote by
.il. qfin, and ail, ...., as: respectively. The demand
functions will be intercorrelated, both among the mines owned
by each competitor and between the mines of different concerns.
Consequently the m integrals Ji representing the discounted prof-
its will involve in their integrands fi all the qij, as well as some at
least of the cumulated productions xi= fqij dt. If the ith owner
wishes to make his profit a maximum, assuming the production
rates of the others to have been fixed upon, he will adjust his ni
production rates so that
O0if d aO * U (j=I, 2, . , n)
Oxii1 dt 0qi1
Continuing the analogy with the static case, we are to imagine
that the other competitors, hearing of his plans, do likewise,
altering their schedules to conform to equations resembling those
above. When the ith owner learns of their changed plans, he will
in turn readjust. The only possible final equilibrium with a set-
tled schedule of production for each mine will be determined by
the solution of the set of differential equations of this type, which
are exactly as numerous as the mines, and therefore as the vari'3 Harold Hotelling, "Stability in Competition," Economic Journal, XLI (March,
I929), 4I.
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES I73
ables to be determined. All this is a direct generalization of the
case of inexhaustible supplies. But we shall show that the solution tends to overstate the production rates and understate the
prices of competing mines.
Doubts in plenty have been cast upon the result in the simpler
case, and the reasons which can there be adduced in favor of the
solution are even more painfully inadequate when the supplies
are of limited amount. The chief difficulty with the problem of a
small number of sellers consists in the fact that each, in modify-
ing his conduct in accordance with what he thinks the others are
going to do, may or may not take account of the effect upon their
prices and policies of his own prospective acts. There is an "equilibrium point," such that neither of two sellers can, by changing
his price, increase his rate of profit while the other's price remains
unchanged. However, if one seller increases his price moderately,
thus making some immediate sacrifice, the other will find his
most profitable course to lie in increasing his own price; and then,
if the original increase is not too great, both will obtain higher
profits than at "equilibrium." But that the tendency to cut prices
below the equilibrium is less important than has been supposed is
shown in the article just referred to.
With an exhaustible supply, and therefore with less to lose by
a temporary reduction in sales, a seller will be particularly inclined to experiment by raising his price above the theoretical
level in the hope that his competitors will also increase their
prices. For the loss of business incurred while waiting for them
to do so he can in this case take comfort, not merely in the
prospect of approximating his old sales at the higher price in the
near future but also in the fact that he is conserving his supplies
for a time when general exhaustion will be nearer and even the
theoretical price will be higher. Thus a general condition may be
expected of higher prices and lower rates of production than are
given by the solution of the simultaneous characteristic equations.
For complementary products, such as iron and coal, the situation is in some ways reversed. Edgeworth in his Papers Relating
to Political Economy points out that when two complementary
goods are separately monopolized the consumer is worse off than
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I74 HAROLD HOTELLING
if both were under the control of the same monopolist. This assumes the equilibrium solution to hold. The tentative deviations
from equilibrium made in order to influence the other party may
now be in either direction, according as the nature of the demand
function and other conditions make it more profitable to move
toward the lower prices and larger sales characterizing the maximum joint profit, or to raise one's price in an effort to force one's
rival to lower his in order to maintain sales. When the supplies
of the complementary goods are exhaustible, the same indeterminateness exists.
A very different problem of duopoly involving the calculus of
variations has been studied by C. F. RoosI4 who finds'5 that the
respective profits take true maximum values. However, as in the
static case, no definitively stable equilibrium is insured by the
fact that each profit is a maximum when the other is considered
fixed, since the acts of one competitor affect those of the other.
The calculus of variations is used by Roos and G. C. Evans'6 to
deal with cost and demand functions involving the rate of change
of price as well as the price. Such functions we have for concreteness and simplicity avoided, but if they should prove to be
of importance in mine economics the foregoing treatment can
readily be extended to them. (Cf. ? I2.) Evans and Roos are not
concerned with exhaustible assets, and assume that at any time
all competitors sell at the same price.
The problems of exhaustible resources involve the time in an-
other way besides bringing on exhaustion and higher prices, namely, as bringing increased information, both as to the physical extent and condition of the resource and as to the economic phe-
nomena attending its extraction and sale. In the most elemen-
tary discussions of exchange, as in bartering nuts for apples, as
well as in discussions of duopoly, a time element is always intro-
duced to show a gradual approach to equilibrium or a breaking
14 "A Mathematical Theory of Competition," American Journal of Mathematics,
XLVII (1925), i63.
'I "Generalized Lagrange Problems in the Calculus of Variations," Transactions
of the American Mathematical Society, XXX (I927), 360.
I6 A Mathematical Introduction to Economics (McGraw-Hill, I930).
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES 175
away from it. Such time effects are equally or in even greater
measure involved in exploiting irreplaceable assets, entangling
with the secular tendencies peculiar to this class of goods. With
duopoly in the sale of exhaustible resources the possibilities of
bargaining, bluff, and bluster become remarkably intricate.
The periodic price wars which break the monotony of gasoline
prices on the American Pacific Coast are an interesting phenomenon. Along most of the fifteen-hundred-mile strip west of the
summits of the Sierras a few large companies dominate the oil
business. In the southern California oil fields, however, numerous small concerns sell gasoline at cut prices. Cheap gasoline is
for the most part not distilled from oil but is filtered from natural
gas, and may be of slightly inferior quality; nevertheless, it is an
acceptable motor fuel. The extreme mobility of purchasers of
gasoline reduces to a minimum the element of gradualness in the
shift of demand from seller to seller with change of price. Ordinarily, the price outside of southern California is held steady by
agreement among the five or six major companies, being fixed
in each of several large areas according to distance from the oil
fields. But every year or two a price war occurs, in which prices
go down day by day to extremely low levels, sometimes almost to
the point of giving away gasoline, and certainly below the cost of
distribution. From a normal price of 20 to 23 cents a gallon the
price sometimes drops to 6 or 7 cents, including the tax of 3 cents.
Peace is made and the old high price restored after a few weeks of
universal joy-riding and storage in every available container,
even in bath tubs. The interesting thing is the slowness of the
spread of these contests, which usually begin in southern California. The companies fight each other violently there, and a few
weeks later in northern California, while in some cases maintaining full prices in Oregon and Washington. These affrays give an
example of the instability of competition when variations of price
with location as well as time complicate commerce in an exhaustible asset.
HAROLD HOTELLING
STANFORD UNIVERSITY
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